Projecting the Fokker-Planck Equation onto a finite dimensional exponential family

TL;DR

This paper develops a rigorous method for projecting the evolution of diffusion process densities onto finite-dimensional exponential families within the statistical exponential manifold framework, with results on compactness and future convergence analysis.

Contribution

It introduces a formal approach to project diffusion densities onto finite-dimensional exponential manifolds using the infinite-dimensional exponential manifold structure.

Findings

Projection interpreted as evolution of a different diffusion process

Compactness result as dimension increases

Foundation for future convergence studies

Abstract

In the present paper we discuss problems concerning evolutions of densities related to Ito diffusions in the framework of the statistical exponential manifold. We develop a rigorous approach to the problem, and we particularize it to the orthogonal projection of the evolution of the density of a diffusion process onto a finite dimensional exponential manifold. It has been shown by D. Brigo (1996) that the projected evolution can always be interpreted as the evolution of the density of a different diffusion process. We give also a compactness result when the dimension of the exponential family increases, as a first step towards a convergence result to be investigated in the future. The infinite dimensional exponential manifold structure introduced by G. Pistone and C. Sempi is used and some examples are given.